Fermi Problems — How to Think like a Physicist [How many workers are necessary to build Khufu’s pyramid?]

“For the making of the pyramid itself there passed a period of twenty years. They worked by a hundred thousand men at a time, for each three months continually.”

Herodotus (484 – c. 425 BC)

Pyramids of Giza, soaring above the city of Cairo, Egypt 

March 1st 2022 — A ‘Fermi Question’ asks for a quick estimate of a quantity that seems difficult, or even impossible, to determine precisely. Fermi’s approach to such questions was to use common sense and rough estimates of quantities to piece together a ball-park value. Follow us on Twitter: @INTEL_TODAY

RELATED POST: CIA Dr Richards J. Heuer: “Information Collection vs Analytical Methods”

Italian physicist Enrico Fermi (1901-1954) received the 1938 Nobel Prize for his discovery of nuclear reactions caused by slow neutrons. Fermi was famous for being able to make surprisingly good estimates in situations where very little information was known.

Intelligence analysts could learn a great deal from solving this kind of problems. Yet, I am aware of only one case where the CIA has actually used this method.

Curious? Let’s give it a try… How many workers are necessary to build Khufu’s pyramid?

Modern egyptologists estimate a workforce of about 20’000 to 25’000 men. [1] Here is a rough estimate based on the minimum of workers necessary to finish the pyramid in 20 years time, which is a time constraint provided by the length of Khufu’s reign.

Known facts The Great Pyramid of Cheops was built more than 4500 years ago. The height was about 146.5 metres, and the base was measured to be about 230.3 metres square, giving a volume of roughly 2.6 million cubic metres.

Assuming a density of 2.6 metric tons per cubic meter, we obtain a total mass of about 6.75 million metric tons.

Potential Energy The potential energy of the pyramid — the energy needed to lift the mass above ground level — is simply the product of acceleration due to gravity, mass, and the center of mass, which in a pyramid is one-quarter of its height. [In the International System (kg, m, s), the unit of energy/work is the Joule.]

Therefore, the potential energy of the Great pyramid is about 2.4 trillion joules. (2.4 × 1012 J)

How much work can you deliver daily? This one is a bit tricky, but I will accept the guesstimate of Vaclav Smil. [2]

To maintain his basal metabolic rate, a 70-kilogram (154-pound) man requires some 7.5 mega-joules a day; steady exertion will raise that figure by at least 30 percent. About 20 percent of that increase will be converted into useful work, which amounts to about 450 kilo-joules a day.

For those who like to bench-press at the gym… 20 repetitions of 50 kg mounts to about 10 kilo-joules. Thus, you would have to perform 45 such series to deliver 450 kilo-joules in a day. That is hard work!

How many man-days to raise the pyramid? Dividing the potential energy of the pyramid by 450 kJ implies that it took 5.4 million man-days to raise the pyramid.

Assuming that the workers had a weekly day off and a few holidays, the work year consisted of about 300 days. This amounts to 18,000 man-years, which — spread over 20 years — implies a workforce of about 900 men.

Of course, only a fraction of the workers lifted blocks. Most were involved in support services such as building ramps, bringing food and water, cutting and hauling blocks to the site. The total number of workers was probably around ten times the number of block lifters.

And thus we conclude that about 10,000 workers built Khufu’s pyramid over a 20 years period. This ball-park value fits amazingly well with a rather recent archaeological discovery.

In the 1990s, archaeologists uncovered a cemetery for workers and the foundations of a settlement used to house the builders of the two later pyramids at the site, indicating that no more than 20,000 people lived there.

PS — The Manhattan Project began modestly in 1939, but grew to employ more than 130,000 people and cost nearly US$2 billion (equivalent to about $23 billion in 2020). Question — How much do you think it would cost to build the Great Pyramid today?

“Do You Really Need More Information? The US Intelligence Community invests heavily in improved intelligence collection systems while managers of analysis lament the comparatively small sums devoted to enhancing analytical resources, improving analytical methods, or gaining better understanding of the cognitive processes involved in making analytical judgments.”

Psychology of Intelligence Analysis — Richards J. Heuer


1) How many workers are necessary to build Khufu’s pyramid?

The detailed calculations about how many men were needed to quarry the stones, transport them up the pyramid of Khufu are based on calculations by Dr. Heribert Illig and Franz Löhner



Fermi Problems — How to Think like a Physicist

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